Borel globalizations of partial actions of Polish groups

We show that the enveloping space ({mathbb {X}}_G) of a partial action of a Polish group G on a Polish space ({mathbb {X}}) is a standard Borel space, that is to say, there is a topology (tau ) on ({mathbb {X}}_G) such that (({mathbb {X}}_G, tau )) is Polish and the quotient Borel structure on ({mathbb {X}}_G) is equal to (Borel({mathbb {X}}_G,tau )). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group action and also show that some properties of the Vaught’s transform are valid for partial actions of groups.